IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v87y2023i2d10.1007_s10898-022-01235-y.html
   My bibliography  Save this article

A note on the SDP relaxation of the minimum cut problem

Author

Listed:
  • Hao Hu

    (Clemson University)

  • Xinxin Li

    (Jilin University)

  • Jiageng Wu

    (Jilin University)

Abstract

In a recent paper by Li et al. (Comput Optim Appl 78(3):853–891, 2021), the authors propose an SDP relaxation for solving the minimum cut problem. In this note, we derive this SDP relaxation by Li et al. in a different way which provides a new perspective. Moreover, we give a rigorous proof that this relaxation by Li et al. is a stronger relaxation than the SDP relaxation presented in Pong et al. (Comput Optim Appl 63(2):333–364, 2016). The technique that we use to compare these two relaxations is through establishing a containment relationship between the feasible sets of these two relaxations. This technique is then studied and analyzed in a more general setting. We present some interesting observations about the differences in the optimal sets and values under this technique. Finally, we verify the strength of the SDP relaxation by Li et al. on some random graphs and also on some classical graphs for solving the vertex separator problems in the numerical experiments.

Suggested Citation

  • Hao Hu & Xinxin Li & Jiageng Wu, 2023. "A note on the SDP relaxation of the minimum cut problem," Journal of Global Optimization, Springer, vol. 87(2), pages 857-876, November.
    • Handle: RePEc:spr:jglopt:v:87:y:2023:i:2:d:10.1007_s10898-022-01235-y
    DOI: 10.1007/s10898-022-01235-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-022-01235-y
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10898-022-01235-y?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Ting Pong & Hao Sun & Ningchuan Wang & Henry Wolkowicz, 2016. "Eigenvalue, quadratic programming, and semidefinite programming relaxations for a cut minimization problem," Computational Optimization and Applications, Springer, vol. 63(2), pages 333-364, March.
    2. William W. Hager & James T. Hungerford & Ilya Safro, 2018. "A multilevel bilinear programming algorithm for the vertex separator problem," Computational Optimization and Applications, Springer, vol. 69(1), pages 189-223, January.
    3. Fanz Rendl & Renata Sotirov, 2018. "The min-cut and vertex separator problem," Computational Optimization and Applications, Springer, vol. 69(1), pages 159-187, January.
    4. Qing Zhao & Stefan E. Karisch & Franz Rendl & Henry Wolkowicz, 1998. "Semidefinite Programming Relaxations for the Quadratic Assignment Problem," Journal of Combinatorial Optimization, Springer, vol. 2(1), pages 71-109, March.
    5. Naomi Graham & Hao Hu & Jiyoung Im & Xinxin Li & Henry Wolkowicz, 2022. "A Restricted Dual Peaceman-Rachford Splitting Method for a Strengthened DNN Relaxation for QAP," INFORMS Journal on Computing, INFORMS, vol. 34(4), pages 2125-2143, July.
    6. Xinxin Li & Ting Kei Pong & Hao Sun & Henry Wolkowicz, 2021. "A strictly contractive Peaceman-Rachford splitting method for the doubly nonnegative relaxation of the minimum cut problem," Computational Optimization and Applications, Springer, vol. 78(3), pages 853-891, April.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Hu, Hao & Sotirov, Renata & Wolkowicz, Henry, 2023. "Facial reduction for symmetry reduced semidefinite and doubly nonnegative programs," Other publications TiSEM 8dd3dbae-58fd-4238-b786-e, Tilburg University, School of Economics and Management.
    2. Xinxin Li & Ting Kei Pong & Hao Sun & Henry Wolkowicz, 2021. "A strictly contractive Peaceman-Rachford splitting method for the doubly nonnegative relaxation of the minimum cut problem," Computational Optimization and Applications, Springer, vol. 78(3), pages 853-891, April.
    3. Norberto Castillo-García & Paula Hernández Hernández, 2019. "Two new integer linear programming formulations for the vertex bisection problem," Computational Optimization and Applications, Springer, vol. 74(3), pages 895-918, December.
    4. Naomi Graham & Hao Hu & Jiyoung Im & Xinxin Li & Henry Wolkowicz, 2022. "A Restricted Dual Peaceman-Rachford Splitting Method for a Strengthened DNN Relaxation for QAP," INFORMS Journal on Computing, INFORMS, vol. 34(4), pages 2125-2143, July.
    5. Fanz Rendl & Renata Sotirov, 2018. "The min-cut and vertex separator problem," Computational Optimization and Applications, Springer, vol. 69(1), pages 159-187, January.
    6. Samuel Burer & Kyungchan Park, 2024. "A Strengthened SDP Relaxation for Quadratic Optimization Over the Stiefel Manifold," Journal of Optimization Theory and Applications, Springer, vol. 202(1), pages 320-339, July.
    7. Lennart Sinjorgo & Renata Sotirov, 2024. "On Solving MAX-SAT Using Sum of Squares," INFORMS Journal on Computing, INFORMS, vol. 36(2), pages 417-433, March.
    8. Ting Pong & Hao Sun & Ningchuan Wang & Henry Wolkowicz, 2016. "Eigenvalue, quadratic programming, and semidefinite programming relaxations for a cut minimization problem," Computational Optimization and Applications, Springer, vol. 63(2), pages 333-364, March.
    9. Godai Azuma & Mituhiro Fukuda & Sunyoung Kim & Makoto Yamashita, 2023. "Exact SDP relaxations for quadratic programs with bipartite graph structures," Journal of Global Optimization, Springer, vol. 86(3), pages 671-691, July.
    10. Jiming Peng & Tao Zhu & Hezhi Luo & Kim-Chuan Toh, 2015. "Semi-definite programming relaxation of quadratic assignment problems based on nonredundant matrix splitting," Computational Optimization and Applications, Springer, vol. 60(1), pages 171-198, January.
    11. Yichuan Ding & Dongdong Ge & Henry Wolkowicz, 2011. "On Equivalence of Semidefinite Relaxations for Quadratic Matrix Programming," Mathematics of Operations Research, INFORMS, vol. 36(1), pages 88-104, February.
    12. de Klerk, E. & Sotirov, R., 2007. "Exploiting Group Symmetry in Semidefinite Programming Relaxations of the Quadratic Assignment Problem," Other publications TiSEM 87a5d126-86e5-4863-8ea5-1, Tilburg University, School of Economics and Management.
    13. Yong Xia & Wajeb Gharibi, 2015. "On improving convex quadratic programming relaxation for the quadratic assignment problem," Journal of Combinatorial Optimization, Springer, vol. 30(3), pages 647-667, October.
    14. Sungwoo Park & Dianne P. O’Leary, 2015. "A Polynomial Time Constraint-Reduced Algorithm for Semidefinite Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 166(2), pages 558-571, August.
    15. Klerk, Etienne de, 2010. "Exploiting special structure in semidefinite programming: A survey of theory and applications," European Journal of Operational Research, Elsevier, vol. 201(1), pages 1-10, February.
    16. F. Rendl, 2016. "Semidefinite relaxations for partitioning, assignment and ordering problems," Annals of Operations Research, Springer, vol. 240(1), pages 119-140, May.
    17. E. de Klerk & R. Sotirov & U. Truetsch, 2015. "A New Semidefinite Programming Relaxation for the Quadratic Assignment Problem and Its Computational Perspectives," INFORMS Journal on Computing, INFORMS, vol. 27(2), pages 378-391, May.
    18. Levent Tunçel & Henry Wolkowicz, 2012. "Strong duality and minimal representations for cone optimization," Computational Optimization and Applications, Springer, vol. 53(2), pages 619-648, October.
    19. Renata Sotirov, 2014. "An Efficient Semidefinite Programming Relaxation for the Graph Partition Problem," INFORMS Journal on Computing, INFORMS, vol. 26(1), pages 16-30, February.
    20. Kuryatnikova, Olga & Sotirov, Renata & Vera, J.C., 2022. "The maximum $k$-colorable subgraph problem and related problems," Other publications TiSEM 40e477c0-a78e-4ee1-92de-8, Tilburg University, School of Economics and Management.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jglopt:v:87:y:2023:i:2:d:10.1007_s10898-022-01235-y. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.